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Contrasts in ANOVA. Decomposition of Treatment Sums of Squares using prior information on the structure of the treatments and/or treatment groups. Contrasts, notation….

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Contrasts in anova

Contrasts in ANOVA

Decomposition of Treatment Sums of Squares using prior information on the structure of the treatments and/or treatment groups

Contrasts notation
Contrasts, notation….

For a Oneway ANOVA, a contrast is a specific comparison of Treatment group means. Contrast constants are composed to test a specific hypothesis related to Treatment means based upon some prior information about the Treatment groups. For k treatment groups, contrast constants are a sequence of numbers

such that

Contrasts and hypothesis testing
Contrasts and Hypothesis testing

A given contrast will test a specific set of hypotheses:



to create an F-statistic with one numerator df.

Example 1 control and two equivalent treatments
Example 1: Control and two equivalent treatments

Suppose we have two treatments which are supposed to be equivalent. For example, each of two drugs is supposed to work by binding to the receptor for adrenalin. Propanolol is such a drug sometimes used for hypertension or anxiety.

We may think that:

  • the two drugs are equivalent, and

  • they are different from Control

The two contrasts
The two contrasts:

Control Drug A Drug B

Contrast 1 -1 ½ ½

Contrast 2 0 -1 +1

Contrast 1 tests whether or not the Control group differs from the groups which block the adrenalin receptors.

Contrast 2 tests whether or not the two drugs differ in their effect.

Orthogonal contrasts
Orthogonal Contrasts

  • The contrasts in the last example were orthogonal.

  • Two contrasts are orthogonal if the pairwise products of the terms sum to zero.

  • The formal definition is that two contrasts


    are orthogonal if:

Orthogonal contrasts allow the trt sums of squares to be decomposed
Orthogonal Contrasts allow the Trt Sums of Squares to be decomposed

The Trt Sums of Squares can be written as a sum of two Statistically independent terms:

Which can be used to test the hypotheses in the example. The a priori structure in the Treatments can be tested for significance in a more powerful way.


If all of the differences in the means are described by one of the contrasts, say the first contrast, then

is more likely to be significant than

Since the signal in the numerator is not combined with “noise”.

Because there is structure to the treatment groups involving drugs and gender
Because there is structure to the Treatment groups involving Drugs and Gender

We can look into the Main Effects of Drug and Gender and Interaction via Orthogonal Contrasts

Drug A A B B

Gender M F M F

Contrast 1 +1/2 +1/2 -1/2 -1/2

Contrast 2 +1/2 -1/2 +1/2 -1/2

Contrast 3 +1/2 -1/2 -1/2 +1/2

The contrasts correspond to the main effects and interaction terms
The Contrasts correspond to the Main Effects and Interaction terms

  • Contrast 1 is the Main effect for Drug

  • Contrast 2 is the Main effect for Gender

  • Contrast 3 is the Interaction term

  • The Sums of Squares for these Contrasts adds up to the Sums of Squares Model in the Two-way ANOVA since each pair of Contrasts is orthogonal